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They were first studied by Bochner [1] and Stein [11], [12], [13] in connection with summation of multiple Fourier Series. If X is bigger than a critical exponent depending of the dimension (X > (n -1)/2), then the kernel of TX is integrable and therefore TX is bounded on every LP(Rn). Stein [11] and Calderon and Zygmund [2], showed that TA is bounded on LP, 1 < p < oo whenX=critical index=(n-1)/2.
The problem then, arises when we consider X smaller than that critical index. Herz [9] (See also Fefferman [4]) pointed out that TX is unbounded outside the range p (X)2 n +1+ < P < n -2S-n (X).n+I+2/X n -I-2X-
Fefferman [5] showed that T is never bounded on L P except for the obvious cases: n = 1 or p =2; and also Fefferman [4] proved that TX is bounded on L P (Rn) provided that p (X) < p < p'(X) and X > (n -1)/4. This result has been sharpened by Tomas [15] to X > (n -1)/2(n + 1). Finally Carleson and Sjolin [3], Fefferman [6] and Hormander [10] proved that, in R2, TA is bounded on LP whenever X >0 and p(X)
2 we have the natural question: is TX (X >0) bounded on L P(R'), p(X) < p < p'(X)?Our approach to the problem is inspired by the work of Fefferman and it is as follows: The multiplier theorem for TX can be easily reduced to this problem:
We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.
We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy's law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two-dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.
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